arXiv subjects

Michela Guida

Publications and source records attributed to Michela Guida.

11 recordsLinked to original sources

arXiv2022

A note on quasilinear Schrödinger equations with singular or vanishing radial potentials

In this note we complete a previous study, where we got existence results for the quasilinear elliptic equation \begin{equation*} -Δw+ V\left( \left| x\right| \right) w - w \left( Δw^2 \right)= K(|x|) g(w) \quad \text{in }\mathbb{R}^{N}, \end{equation*} with singular or vanishing continuous radial potentials $V(r)$, $K(r)$. In our previuos study we assumed, for technical reasons, that $K(r)$ was vanishing as $r \rightarrow 0$, while in the present paper we remove this obstruction. To face the problem we apply a suitable change of variables $w=f(u)$ and we find existence of non negative solutions by the application of variational methods. Our solutions satisfy a weak formulations of the above equation, but they are in fact classical solutions in $\mathbb{R}^{N} \setminus \{0\}$. The nonlinearity $g$ has a double-power behavior, whose standard example is $g(t) = \min \{ t^{q_1 -1}, t^{q_2 -1} \}$ ($t>0$), recovering the usual case of a single-power behavior when $q_1 = q_2$.

math.AP↗

arXiv2022

Existence results for a class of quasilinear Schrödinger equations with singular or vanishing potentials

Given two continuous functions $V\left(r \right)\geq 0$ and $K\left(r\right)> 0$ ($r>0$), which may be singular or vanishing at zero as well as at infinity, we study the quasilinear elliptic equation \[ -Δw+ V\left( \left| x\right| \right) w - w \left( Δw^2 \right)= K(|x|) g(w) \quad \text{in }\mathbb{R}^{N}, \] where $N\geq3$. To study this problem we apply a change of variables $w=f(u)$, already used by several authors, and find existence results for nonnegative solutions by the application of variational methods. The main features of our results are that they do not require any compatibility between how the potentials $V$ and $K$ behave at the origin and at infinity, and that they essentially rely on power type estimates of the relative growth of $V$ and $K$, not of the potentials separately. Our solutions satisfy a weak formulations of the above equation, but we are able to prove that they are in fact classical solutions in $\mathbb{R}^{N} \backslash \{ 0\}$. To apply variational methods, we have to study the compactness of the embedding of a suitable function space into the sum of Lebesgue spaces $L_{K}^{q_{1}}+L_{K}^{q_{2}}$, and thus into $L_{K}^{q}$ ($=L_{K}^{q}+L_{K}^{q}$) as a particular case. The nonlinearity $g$ has a double-power behavior, whose standard example is $g(t) = \min \{ t^{q_1 -1}, t^{q_2 -1} \}$, recovering the usual case of a single-power behavior when $q_1 = q_2$.

math.AP↗

arXiv2022

Radial quasilinear elliptic problems with singular or vanishing potentials

In this paper we continue the work that we began in arXiv:1912.07537. Given $1 0$, and a continuous function $A(r) >0\ (r>0)$, we consider the quasilinear elliptic equation \[ -\mathrm{div}\left(A(|x| )|\nabla u|^{p-2} \nabla u\right) +V\left( \left| x\right| \right) |u|^{p-2}u= K(|x|) f(u) \quad \text{in }\mathbb{R}^{N}, \] where all the potentials $A,V,K$ may be singular or vanishing, at the origin or at infinity. We find existence of nonnegative solutions by the application of variational methods, for which we need to study the compactness of the embedding of a suitable function space $X$ into the sum of Lebesgue spaces $L_{K}^{q_{1}}+L_{K}^{q_{2}}$. The nonlinearity has a double-power super $p$-linear behavior, as $f(t)= \min \left\{ t^{q_1 -1}, t^{q_2 -1} \right\}$ with $q_1,q_2>p$ (recovering the power case if $q_1=q_2$). With respect to \cite{AVK_I}, in the present paper we assume some more hypotheses on $V$, and we are able to enlarge the set of values $q_1 , q_2$ for which we get existence results.

math.AP↗

arXiv2019

Compactness and existence results for quasilinear elliptic problems with singular or vanishing potentials

Given $N\geq 3$, $1 0$ and a continuous function $A(r) >0$ ($r>0$), we study the quasilinear elliptic equation \[ -\mathrm{div}\left(A(|x| )|\nabla u|^{p-2} \nabla u\right) u+V\left( \left| x\right| \right) |u|^{p-2}u= K(|x|) f(u) \quad \text{in }\mathbb{R}^{N}. \] We find existence of nonegative solutions by the application of variational methods, for which we have to study the compactness of the embedding of a suitable function space $X$ into the sum of Lebesgue spaces $L_{K}^{q_{1}}+L_{K}^{q_{2}}$, and thus into $L_{K}^{q}$ ($=L_{K}^{q}+L_{K}^{q}$) as a particular case. Our results do not require any compatibility between how the potentials $A$, $V$ and $K$ behave at the origin and at infinity, and essentially rely on power type estimates of the relative growth of $V$ and $K$, not of the potentials separately. The nonlinearity $f$ has a double-power behavior, whose standard example is $f(t) = \min \{ t^{q_1 -1}, t^{q_2 -1} \}$, recovering the usual case of a single-power behavior when $q_1 = q_2$.

math.AP↗

arXiv2016

Compactness and existence results for the $p$-Laplace equation

Given $1 0$, $r>0$, we define the weighted spaces \[ W=\left\{ u\in D^{1,p}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}V\left( \left| x\right| \right) \left| u\right| ^{p}dx<\infty \right\} ,\quad L_{K}^{q}=L^{q}(\mathbb{R}^{N},K\left( \left| x\right| \right) dx) \] and study the compact embeddings of the radial subspace of $W$ into $L_{K}^{q_{1}}+L_{K}^{q_{2}}$, and thus into $L_{K}^{q}$ ($=L_{K}^{q}+L_{K}^{q}$) as a particular case. We consider exponents $q_{1},q_{2},q$ that can be greater or smaller than $p$. Our results do not require any compatibility between how the potentials $V$ and $K$ behave at the origin and at infinity, and essentially rely on power type estimates of their relative growth, not of the potentials separately. We then apply these results to the investigation of existence and multiplicity of finite energy solutions to nonlinear $p$-Laplace equations of the form \[ -\triangle _{p}u+V\left( \left| x\right| \right) |u|^{p-1}u=g\left( \left| x\right| ,u\right) \quad \text{in }\mathbb{R}^{N},\ 1<p<N, \] where $V$ and $g\left( \left| \cdot \right| ,u\right) $ with $u$ fixed may be vanishing or unbounded at zero or at infinity. Both the cases of $g$ super and sub $p$-linear in $u$ are studied and, in the sub $p$-linear case, nonlinearities with $g\left( \left| \cdot \right| ,0\right) \neq 0$ are also considered.

math.AP↗

arXiv2015

Compactness results for the $p$-Laplace equation

Given $1 0$, $r>0$, we define the weighted spaces \[ W=\left\{ u\in D^{1,p}(\mathbb{R}^N):\int_{\mathbb{R}^N}V\left(\left|x\right|\right) \left|u\right|^p dx<\infty \right\} , \quad L_{K}^q =L^q(\mathbb{R}^N,K\left( \left| x\right| \right) dx) \] and study the compact embeddings of the radial subspace of $W$ into $L_{K}^{q_1}+L_{K}^{q_2}$, and thus into $L_{K}^q$ ($=L_{K}^q+L_{K}^q$) as a particular case. Both exponents $q_1,q_2,q$ greater and lower than $p$ are considered. Our results do not require any compatibility between how the potentials $V$ and $K$ behave at the origin and at infinity, and essentially rely on power type estimates of their relative growth, not of the potentials separately.

math.AP↗

arXiv2015

Compactness and existence results in weighted Sobolev spaces of radial functions. Part II: Existence

We prove existence and multiplicity results for finite energy solutions to the nonlinear elliptic equation \[ -\triangle u+V\left( \left| x\right| \right) u=g\left( \left| x\right| ,u\right) \quad \textrm{in }Ω\subseteq \mathbb{R}^{N},\ N\geq 3, \] where $Ω$ is a radial domain (bounded or unbounded) and $u$ satisfies $u=0$ on $\partial Ω$ if $Ω\neq \mathbb{R}^{N}$ and $u\rightarrow 0$ as $\left| x\right| \rightarrow \infty $ if $Ω$ is unbounded. The potential $V$ may be vanishing or unbounded at zero or at infinity and the nonlinearity $g$ may be superlinear or sublinear. If $g$ is sublinear, the case with $g\left( \left| \cdot \right| ,0\right) \neq 0$ is also considered.

math.AP↗

arXiv2015

Nonlinear Schrödinger equations without compatibility conditions on the potentials

We study the existence of nonnegative solutions (and ground states) to the nonlinear Schrödinger equation in $\mathbb{R}^N$ with radial potentials and super-linear or sub-linear nonlinearities. The potentials satisfy power type estimates at the origin and at infinity, but no compatibility condition is required on their growth (or decay) rates at zero and infinity. In this respect our results extend some well known results in the literature and we also believe that they can highlight the role of the sum of Lebesgue spaces in studying nonlinear equations with weights.

math.AP↗

arXiv2014

Some abstract results on the existence of bounded Palais-Smale sequences

Without compactness assumptions, we prove some abstract results which show that a $C^{1}$ functional $I:X\rightarrow \mathbb{R}$ on a Banach space $X$ admits bounded Palais-Smale sequences provided that it exhibits some geometric structure of minimax type and a suitable behaviour with respect to some sequence of continuous mappings $ψ_{n}:X\rightarrow X$. This work is a preliminary version of a forthcoming paper, where applications to nonlinear equations without Ambrosetti-Rabinowitz type assumptions will also be given.

math.FA↗

arXiv2014

Compactness and existence results in weighted Sobolev spaces of radial functions, Part I: Compactness

Given two measurable functions $V(r)\geq 0$ and $K(r)> 0$, $r>0$, we define the weighted spaces \[ H_V^1 = \{u \in D^{1,2}(\mathbb{R}^N): \int_{\mathbb{R}^N}V(|x|)u^{2}dx < \infty \}, \quad L_K^q = L^q(\mathbb{R}^N,K(|x|)dx) \] and study the compact embeddings of the radial subspace of $H_V^1$ into $L_K^{q_1}+L_K^{q_2}$, and thus into $L_K^q$ ($=L_K^q+L_K^q$) as a particular case. Both super- and sub-quadratic exponents $q_1$, $q_2$ and $q$ are considered. Our results do not require any compatibility between how the potentials $V$ and $K$ behave at the origin and at infinity, and essentially rely on power type estimates of their relative growth, not of the potentials separately. Applications to existence results for nonlinear elliptic problems like \[ -\triangle u + V(|x|)u = f(|x|,u) \quad \text{in}\mathbb{R}^N, \quad u \in H_V^1, \] will be given in a forthcoming paper.

math.FA↗

arXiv2013

A nonexistence result for a nonlinear elliptic equation with singular and decaying potential

The paper deals with positive radial solutions to a nonlinear elliptic equation with singular and decaying potential, for which several existence and nonexistence results are known, resting upon suitable compatibility conditions between the decaying rate of the potential and the growth rate of the nonlinearity. The problem of the existence is still open for essentially three cases and we give a negative answer to one of such cases.

math.AP↗